Question

If y = (cot⁻¹ x)², prove that y₂(x² + 1)² + 2x (x² + 1) y₁ = 2.

Answer 1

Here,
$$\begin{gathered} y={\left({\cot }^{-1}x\right)}^2 \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ {y}_1=2\mathrm{c}\mathrm{o}{\mathrm{t}}^{-1}x\times \frac{-1}{1+x^2}=\frac{-2\mathrm{c}\mathrm{o}{\mathrm{t}}^{-1}x}{1+x^2} \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ {y}_2=\frac{2+4x{\cot }^{-1}x}{{\left(1+x^2\right)}^2} \\ \Rightarrow y_2=\frac{2}{{\left(1+x^2\right)}^2}+\frac{2x\times 2{\cot }^{-1}x}{\left(1+x^2\right)\left(1+x^2\right)} \\ \Rightarrow y_2=\frac{2}{{\left(1+x^2\right)}^2}-\frac{2x{y}_1}{\left(1+x^2\right)} \\ \Rightarrow {\left(1+x^2\right)}^2{y}_2=2-2x{y}_1\left(1+x^2\right) \\ \Rightarrow {\left(1+x^2\right)}^2{y}_2+2x{y}_1\left(1+x^2\right)=2 \end{gathered}$$

Hence proved.